The Classical Braess Paradox Problem Revisited: A Generalized Inverse Method on Non-Unique Path Flow Cases

Abstract

The classical Braess paradox problem refers to a user-equilibrium assignment model which all started with Braess's (Unternehmensforschung 12; 258-268, 1968) demonstrated example network. Some variants of Braess paradox and related theories were subsequently developed to detect this paradoxical phenomenon on a general network. In this paper, the authors are devoted to the classical Braess paradox problem involving situations whenever considering new links to be added to a network. Historical literature told us that existing theories for this problem were limited to networks which admit unique path flow solution. A generalized inverse approach is suggested to solve this problem without the assumption of unique path flow solution in this study. The change of equilibrium cost after link additions is derived as a generalized inverse formulation of which solution possesses the non-uniqueness and flow conservation over all perturbed paths. Based on this generalized inverse formulation of the change of equilibrium cost, the authors show that there exists at least one of the O/D pairs, connected by new added routes, such that Braess paradox doesn't (does) occur if the proposed test matrix is positive (negative) semi-definite. The derivations extend existing theories towards the situations when multiple routes are arbitrarily generated after link additions. These new theories deliver prior information to foresee Braess paradox taking place on a class of transportation networks which is more general than before and never reached by existing studies on the indicated classical Braess paradox problem.

Airway network management using Braess’s Paradox

“…Note that in the literature when investigating BP on road traffic networks, researchers have utilised linear cost functions [42][43][44], polynomial cost functions [45] and general monotonous non-decreasing cost functions [46]. We regress the cost function using linear function, polynomial function and cubic function and the results show that they have similar performance.…”

A Transportation Network Paradox: Consideration of Travel Time and Health Damage due to Pollution

“…The authors of [31,32] examined the necessary and sufficient conditions when the Braess paradox occurs for general networks with linear link-cost functions. Reference [33] proved that there is at least one O/D pair connected by a new path so that the Braess paradox does not (does) occur, as the proposed test matrix is positive (negative) and semi-definite. Reference [34] showed that the Braess paradox only occurs when the total travel demand falls within a certain intermediate range of demands.…”

Overcoming the Downs-Thomson Paradox by transit subsidy policies

“…This phenomenon is then analytically formulated by Mogridge et al (1987) and is named the "The Downs-Thomson Paradox (D-T Paradox)" thereafter. Furthermore, many other paradoxes have been studied in the transportation literature, including Braess paradox (Braess, 1968(Braess, , 2005Lin and Lo, 2009;Zhao et al, 2014;Di et al, 2014;Zverovich and Avineri, 2015;Hwang and Cho, 2015;, emission paradox (Nagurney, 2000), capacity paradox (Yang and Bell, 1998), stochastic assignment paradox (Sheffi and Daganzo, 1978;Yao and Chen, 2014), transit assignment paradox (Szeto and Jiang, 2014) and so on. This paper focuses on examining the D-T Paradox introduced by Downs (1962) and Thomson (1977).…”